Some finite double integral involving general class of polynomials, special functions , Aleph-function and multivariable I-function

Some finite double integral involving general class of polynomials, special

functions , Aleph-function and multivariable I-function

1 Teacher in High School , France E-mail : [email protected]

ABSTRACT

The aim of the present document is to evaluate three triple double finite integrals involving general class of polynomials, special functions, Aleph-function and multivariable I-Aleph-function. Importance of our findings lies in the fact that they involve the multivariable I-Aleph-function, which are the sufficiently general in nature and are capable of yielding a large number of simpler and useful results merely by specializing the parameters in them. Further we establish some special cases.

KEYWORDS : I-function of several variables, double integrals, special function, general class of polynomials, Aleph-function, multivariable H-function.

2010 Mathematics Subject Classification. 33C60, 82C31

1.Introduction and preliminaries.

The aim of the present document is to evaluate three triple double finite integrals involving general class of polynomials, special functions, Aleph-function and multivariable I-function defined by Prasad [5]. We will study the particular case concerning the multivariable H-function defined by Srivastava et al [7].

The Aleph- function , introduced by Südland [8] et al , however the notation and complete definition is presented here in the following manner in terms of the Mellin-Barnes type integral :

z = (1.1)

for all different to and

= (1.2)

With : Where ;

For convergence conditions and other details of function , see Südland et al [8].the serie representation of Aleph-function is given by Chaurasia et al [1].

(1.3)

With and is given in (1.2) (1.4)

(1.5)

Where are arbitrary positive integers and the coefficients are arbitrary

constants, real or complex. In the present paper, we use the following notation

(1.6)

The multivariable I-function is defined in term of multiple Mellin-Barnes type integral :

=

(1.8)

The defined integral of the above function, the existence and convergence conditions, see Y,N Prasad [5]. Throughout the present document, we assume that the existence and convergence conditions of the multivariable I-function.

The condition for absolute convergence of multiple Mellin-Barnes type contour (1.9) can be obtained by extension of the corresponding conditions for multivariable H-function given by as :

, where

(1.9)

where

We may establish the the asymptotic expansion in the following convenient form :

,

,

where : and

We will use these following notations in this paper :

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

The multivariable I-function write :

_(1.16)

2 . Required integrals

The following integral ([9],p.33;[2],p.172, eq(27);[3],p.46, eq(5) and [4],p.71] will be required to establish our main results :

a ) (2.1)

where Re(c) > 0, Re(2c-a-b) > -1

provided that

c) (2.3)

where

d) (2.4)

where

3. Main integrals

1)

(3.1)

a)

b)

c)

d)

e)

f)

g)

h) , where is defined by (1.9)

i) where

2

)

(3.2)

Provided that

a)

b)

c)

d)

e)

f)

g) , where is defined by (1.9)

h) where

(3.3)

Provided that

a)

b)

c)

d)

e)

f)

g)

i) where

Proof of (3.1)

We first express the general class of polynomial occuring on the L.H.S of (3.1) in series form with the help of (1.5), the Aleph-function in series form with the help of (1.3) and replace the multivariable I-function by its Mellin-Barnes contour integral with the help of (1.7). Now we interchange the order of summation and integrations, we obtain

L.H.S of (3.1) =

Now, using the result (2.1) and (2.2) to evaluate the x-integral and -integral and reinterpreting the multiple contour integral so obtained in the form of multivariable Aleph-function withe help of (1.7), we obtain the desired result. The result (3.2) and (3.1) can be proved by similar prrofs with the help of integrals given by (2.2), (2.3) and (2.4).

4. Multivariable H-function

If , the multivariable I-function defined by Prasad degenere in multivariable H-function defined by Srivastava et al [7]. We have the following results.

(4.1)

where .

which holds true under the same conditions as needed in (3.1) with

2

)

(4.2)

3)

(4.3)

which holds true under the same conditions as needed in (3.3) with

6.

Conclusion

The I-function of several variables presented in this paper, is quite basic in nature. Therefore , on specializing the parameters of this function, we may obtain various other special functions o several variables such as multivariable H-function ,Fox's H-H-function,, Meijer's G-H-function, Wright's generalized Bessel H-function, Wright's generalized hypergeometric function, MacRobert's E-function, generalized hypergeometric function, Bessel function of first kind, modied Bessel function, Whittaker function, exponential function , binomial function etc. as its special cases, and therefore, various unified integral presentations can be obtained as special cases of our results.

REFERENCES

[1] Chaurasia V.B.L and Singh Y. New generalization of integral equations of fredholm type using the Aleph-function Int. J. of Modern Math. Sci. 9(3), 2014, p 208-220

[2] Erdelyi A. et al. Higher transcendental functions. , Vol 1, McGraw-Hill, New-York (1953).

[4] Mathai A.M. And Saxena R.K. (1973) : Generalized hypergeometric functions with applications in statistics and physical sciences, springer-Verlag, Lecture Notes No348, Heidelberg.

[5] Y.N. Prasad , Multivariable I-function , Vijnana Parishad Anusandhan Patrika 29 ( 1986 ) , page 231-237.

[6] Srivastava H.M. A multilinear generating function for the Konhauser set of biorthogonal polynomials suggested by Laguerre polynomial, Pacific. J. Math. Vol 77(1985), page183-191

[7] H.M. Srivastava And R.Panda. Some expansion theorems and generating relations for the H-function of several

complex variables. Comment. Math. Univ. St. Paul. 24(1975), p.119-137.

[8] Südland, N.; Baumann, B. and Nonnenmacher,T.F., Open problem : who knows about the Aleph-functions? Fract. Calc. Appl. Anal., 1(4) (1998): 401-402.

[9] Vyas V.M. and Rathie K., An integral involving hypergeometric function. The mathematics education 31(1997) page33

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